By now in your academic career it should be very apparent just where
homework exists in the grand scheme of (learning) things. Ideally, you
attend a class where a warm and attentive professor clearly explains
some abstruse concept and a whole raft of facts in some moderately
interactive way. Alas, there are too many facts to fit in short
term/immediate memory and too little time to move most of them
through into long term/working memory before finishing with one and
moving on to the next one. As a consequence, by the end of lecture
you've already forgotten many if not most of the facts, but if you
were paying attention, asked questions as needed, and really cared about
learning the material you would remember a handful of the most
important ones, the ones that made your brief understanding of the
material hang (for a brief shining moment) together.

Studies show that you are only likely to retain anywhere from 5% to
30% of what you are shown in lecture. Clearly this is not enough
to make the information conceptually useful, to learn it. In
order to actually learn, you must stop being a passive recipient
of facts. You must actively develop your understanding, by means
of discussing the material and kicking it around with others, by
using the material in some way, by teaching the material to
peers as you come to understand it.

Medical schools have long been aware of this. In the year 1907, medical
schools had two years of coursework to prepare a student to be a
physician. In the year 2007, they are still two years of
coursework - but the amount of science and medicine that is taught in
those two years has exploded. They have developed the following
mantra to help their students understand the only way the process can
still work:

See one. (E.g. a procedure)

Do one. (with a mentor standing by)

Teach one. (still with a mentor, but a more advanced one still)

That's it! We will use our own version of this same process in this
course. Lecture (seeing it done) is important - it sets the stage for
the learning, but by itself it teaches little. Homework (doing it
yourself) is far more important. This is when you begin to really
learn. Recitation (where you teach each other where you have learned)
is where you solidify this learning by articulating it, working with the
concepts in your mind at a high level to do so.

To help facilitate this process, associated with lecture your professor
gave you an assignment. Amazingly enough, its purpose is not to
torment you or to be the basis of your grade (although it may well do
both). It is to give you some concrete stuff to do while thinking
about the material to be learned, while discussing the material to be
learned, while using the material to be learned to accomplish specific
goals, while teaching some of what you figure out to others who are
sharing this whole experience while being taught by them in turn.

In other words, to learn you must do your homework, ideally at
least partly in a group setting. The only question is: how
should you do it to both finish learning all that stuff you sort-of-got
in lecture and to re-attain the moment(s) of clarity that you then
experienced, until eventually it becomes a permanent characteristic of
your awareness and you know and fully understand it all on
your own?

There are two general steps that need to be iterated to finish
learning anything at all. They are a lot of work. In fact, they are
far more work than (passively) attending lecture, and are more important than attending lecture. You can learn the material with
these steps without ever attending lecture, as long as you have
access to what you need to learn in some media or human form. You in
all probability will never learn it, lecture or not, without
making a few passes through these steps. They are:

Review the whole (typically textbooks and/or notes)

Work on the parts (do homework, use it for something)

(iterate until you thoroughly understand whatever it is you are trying
to learn).

Let's examine these steps.

The first is pretty obvious. You didn't ``get it'' from one lecture.
There was too much material. If you were lucky and well prepared
and blessed with a good instructor, perhaps you grasped some of it
for a moment (and if your instructor was poor or you were
particularly poorly prepared you may not have managed even that) but
what you did momentarily understand is fading, flitting further and
further away with every moment that passes. You need to review the
entire topic, as a whole, as well as all its parts. A set of good
summary notes might contain all the relative factoids, but there are
relations between those factoids - a temporal sequencing,
mathematical derviations connecting them to other things you know, a
topical association with other things that you know. They tell a story, or part of a story, and you need to know that story in broad terms, not try to memorize it word for word.

Reviewing the material should be done in layers, skimming the textbook
and your notes, creating a new set of notes out of the text in
combination with your lecture notes, maybe reading in more detail to
understand some particular point that puzzles you, reworking a few of
the examples presented. Lots of increasingly deep passes through it
(starting with the merest skim-reading or reading a summary of the whole
thing) are much better than trying to work through the whole text
one line at a time and not moving on until you understand it. Many
things you might want to understand will only come clear from things you
are exposed to later, as it is not the case that all knowledge is
ordinal, hierarchical, and derivatory.

You especially do not have to work on memorizing the
content. In fact, it is not desireable to try to memorize content
at this point - you want the big picture first so that facts have
a place to live in your brain. If you build them a house, they'll move
right in without a fuss, where if you try to grasp them one at a time
with no place to put them, they'll (metaphorically) slip away again as
fast as you try to take up the next one. Let's understand this a bit.

Your brain is fabulously efficient at storing information in a compressed associative form. It also tends to remember things that are
important - whatever that means - and forget things that aren't
important to make room for more important stuff. There are lots of
experiments that demonstrate this - the simplest being trying to
memorize a string of ten or so numbers at a glance (more than the 7 one
can typically get into short term memory).

Try memorizing 1357902468 from just the one glance you got reading
this sentence. No fair going back and repeating it to yourself, at
least while looking at it! Now look at it and try to remember it. One
strategy is to just repeat it to yourself until you get it right, but if
you stare at it a while and think, you'll see that it has a very
simple pattern embedded in it.

In fact, this number ``compresses'' to a single two-step rule - all the
odd digits in ascending order followed by all the even digits ditto.
You already know what a ``digit'' is, what odd and even numbers
are, what ascending versus descending order is. You only need to
remember "ascending" and "odd followed by even digits" - everything
else is compressed. You will almost certainly be able to remember the
digit string tomorrow without further rehearsal because of this
rule and the fact that it illustrates an interesting point, where if you
didn't notice the pattern and just memorized it as a string of "random"
numbers, devoid of any meaning, your brain would have little reason to
retain it as it is ``unimportant''. Even if you forget this
particular string, you may well remember the point and use a different string like 1212121212 (five repetitions of 12) to illustrate
the same point when teaching it to someone else. This is fine!
My goal, too, is to teach you this, not some particular patterned
set of numbers neither of us really cares about.

This ability to compress goes far beyond what I can explain or you can
easily imagine. When I play a game of chess, I've forgotten my first
five moves by the time I've made my tenth move. By the time the game
finishes, I have no idea how I got into the mess I'm probably in. A
chess master, on the other hand, can finish the game and
then can recontruct the entire game in order, and can criticize
each move as they do so. In fact, they can probably remember the entire
game they played yesterday, or the one they played last week. They've
built a complex structure of associative memory so that they don't
remember moves the same way you or I do.

On the other hand, I can often remember what mistakes a student of
mine made a week after grading one of their papers. I many not remember
the student's name (no good associative memory there) but I've got
great structures for remembering how to solve or not solve physics
problems and remember only what the student did wrong - I already
know how to do what they did right.

This is the goal of your iterated review process. At first you are
memorizing things the hard way, trying to connect what you learn to very
simple hierarchical concepts such as this step comes before that step.
As you do this over and over again, though, you find that absorbing new
information takes you less and less time, and you remember it much more
easily and for a longer time without additional rehearsal. Sometimes
your brain even outruns the learning process and ``discovers'' a
missing part of the structure before you even read about it! By
reviewing the whole, well-organized structure over and over again, you
gradually build a greatly compressed representation of it in your brain
and tremendously reduce the amount of work required to flesh out that
structure with increasing levels of detail and remember them and be
able to work with them for a long, long time.

Now let's understand the second part of doing homework - working
problems. As you can probably guess on your own at this point, there
are good ways and bad ways to do homework problems. The worst way to do
homework (aside from not doing it at all, which is far too common
a practice and a bad idea if you have any intention of learning
the material) is to do it all in one sitting, right before it is due,
and to never again look at it.

It is left as a homework exercise for the student to work out why
this is a bad idea from the discussion and facts given above. So take
a minute and think about it, then turn the page.

Let's see, doing your homework in a single sitting, working on it just
one time fails to repeat and rehearse the material (essential for
turning short term memory into long term in nearly all cases). It exhausts the neurons in your brain (quite literally - there is
metabolic energy consumed in thinking) as one often ends up working on a
problem far too long in one sitting just to get done. It fails to
incrementally build up in your brain's long term memory the structures upon which the more complex solutions are based, so you have
to constantly go back to the book to get them into short term memory
long enough to get through a problem. Even this simple bit of
repetition does initiate a learning process. Unfortunately, by
not repeating them after this one sitting they soon fade, often without
a discernable trace in long term memory.

Just as was the case with memorizing the number above, the problems
almost invariably are not going to be a matter of random noise.
They have certain key facts and ideas that are the basis of their
solution, and those ideas are used over and over again. There is plenty
of pattern and meaning there for your brain to exploit in information
compression, and it may well be very cool stuff to know and hence
important to you once learned, but it takes time and repetition
and a certain amount of meditation for the ``gestalt'' of it to spring
into your awareness and burn itself into your conceptual memory as
``high order understanding''.

You have to give it this time, and perform the repetitions, while
maintaining an optimistic, philosophical attitude towards the process.
You have to do your best to have fun with it. You don't get
strong by lifting light weights a single time. You get strong lifting
weights repeatedly, starting with light weights to be sure, but then
working up to the heaviest weights you can manage. When you do build up to where you're lifting hundreds of pounds, the fifty
pounds you started with seems light as a feather to you.

As with the body, so with the brain. Repeat broad strokes for the big
picture with increasingly deep and ``heavy'' excursions into the
material to explore it in detail as the overall picture emerges.
Intersperse this with sessions where you work on problems and try
to use the material you've figured out so far. Be sure to discuss it and teach it to others as you go as much as possible,
as articulating what you've figured out to others both uses a different
part of your brain than taking it in (and hence solidifies the memory)
and it helps you articulate the ideas to yourself! This process
will help you learn more, better, faster than you ever have before, and
to have fun doing it!

Your brain is more complicated than you think. You are very likely used
to working hard to try to make it figure things out, but
you've probably observed that this doesn't work very well. A lot of
times you simply cannot ``figure things out'' because your brain
doesn't yet know the key things required to do this, or doesn't ``see''
how those parts you do know fit together. Learning and discovery is
not, alas, ``intentional'' - it is more like trying to get a bird to
light on your hand that flits away the moment you try to grasp it.

People who do really hard crossword puzzles (one form of great brain
exercise) have learned the following. After making a pass through the
puzzle and filling in all the words they can ``get'', and maybe making a
couple of extra passes through thinking hard about ones they can't get
right away, looking for patterns, trying partial guesses, they arrive at
an impasse. If they continue working hard on it, they are unlikely to
make further progress, no matter how long they stare at it.

On the other hand, if they put the puzzle down and do
something else for a while - especially if the something else is go to
bed and sleep - when they come back to the puzzle they often can immediately see a dozen or more words that the day before were
absolutely invisible to them. Sometimes one of the long theme
answers (perhaps 25 characters long) where they have no more than two letters just ``gives up'' - they can simply ``see'' what the
answer must be.

Where do these answers come from? The person has not ``figured them
out'', they have ``recognized'' them. They come all at once, and they
don't come about as the result of a logical sequential process.

Often they come from the person's right brain. The left brain
tries to use logic and simple memory when it works on crosswork puzzles.
This is usually good for some words, but for many of the words there are
many possible answers and without any insight one can't even
recall one of the possibilities. The clues don't suffice to
connect you up to a word. Even as letters get filled in this continues
to be the case, not because you don't know the word (although in
really hard puzzles this can sometimes be the case) but because you
don't know how to recognize the word ``all at once'' from a
cleverly nonlinear clue and a few letters in this context.

The right brain is (to some extent) responsible for insight and
non-linear thinking. It sees patterns, and wholes,
not sequential relations between the parts. It isn't intentional - we
can't ``make'' our right brains figure something out, it is often the
other way around! Working hard on a problem, then ``sleeping on it'' is
actually a great way to develop ``insight'' that lets you solve it
without really working terribly hard after a few tries. It also
utilizes more of your brain - left and right brain, sequential
reasoning and insight, and if you articulate it, or use it, or make
something with your hands, then it exercieses these parts of your brain
as well, strengthening the memory and your understanding still more.
The learning that is associated with this process, and the problem
solving power of the method, is much greater than just working on
a problem linearly the night before it is due until you hack your way
through it using information assembled a part at a time from the book.

The following ``Method of Three Passes'' is a specific strategy
that implements many of the tricks discussed above. It is known to be
effective for learning by means of doing homework (or in a generalized
way, learning anything at all). It is ideal for ``problem oriented
homework'', and will pay off big in learning dividends should you adopt
it, especially when supported by a group oriented recitation with
strong tutorial support and many opportunities for peer
discussion and teaching.